On the Mittag Leffler Bargmann (MLB) transform

TL;DR

This paper introduces a new Segal-Bargmann transform linked to the Mittag Leffler Fock space, explores its connection to the Fourier transform, and extends the framework to quaternionic slice hyperholomorphic functions.

Contribution

It develops the Mittag Leffler Bargmann transform, connects it with Fourier analysis, and extends the theory to quaternionic functions, providing new tools in hyperholomorphic analysis.

Findings

Defined the Mittag Leffler Segal-Bargmann transform

Established its relation to the Fourier transform

Extended results to quaternionic slice hyperholomorphic functions

Abstract

We introduce the Segal-Bargmann transform associated to the Mittag Leffler Fock space and study how it will be connected to the Fourier transform. We will discuss also the counterpart of the creation and annihilation operator in this setting using the Caputo and Liouville operators. Finally, we give an extension of these results to the case of quaternions, in particular in the slice hyperholomorphic setting.